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240                               CHEMICAL ENGINEERING
the size of the openings diminishes and the chance of the average grain being fed to any screen of the battery passing through its apertures consequently diminishes as the size of aperture decreases. Or in other words the capacity is proportional to the size of the. aperture.1
So far as the ordinary shaking screen is concerned cognizance under the theory of the velocity on approach is only taken for such velocities as will allow a particle to fall within the boundaries of an aperture. The stroke of most shaking screens is of such vigor that the grains of the smaller sizes will pass over a number of apertures at too high velocities for a trial in them. The theory is only concerned with the velocities the grains have near the ends of the paths they pursue under the actuating mechanism of the screen. The ordinary shaking screen sacrifices good screen work for capacity when screening fine material.
Only the velocity case will be considered where the grain approaches an aperture at right angles to a side and along a line passing through the center of the side. The
equation of a fall parabola with origin at the edge of the opening is x2 = -- -
where V is velocity or approach in feet per second and g the acceleration of gravity also expressed in feet. The maximum permissible velocity with the position of path assumed is obtained when x equals I  l/2n and y equals l/n. Substituting these values V becomes equal to
When n equals unity the conditions are obtained when the chance of the grain going through the aperture becomes 0, and V becomes equal to
The shortest straight-line path will be one which will bring the grain tangent to two sides of the square at some point in the travel, and the longest when the grain pursues a diagonal of the square.
Interpreting the Chance Law.  In the foot note, page 238, it was stated that the ratio of the inner square of Fig. 13 to the whole square was a measure of the percentage of grains of any size smaller than an aperture which would fall through it. In the tabulation below it is assumed that there are a 100 grains each of the sizes 9, 8, 7, 6, 5, 4, 3, 2, and 1 mm. and that they are introduced to the edge of a screen of 10-mm. square openings, the stock between openings having no sensible thickness. It is reckoned under the ratio given above that in passing the first line of apertures 1 per cent of the 9-mm. grains are removed, 4 per cent of the 8-mm., 9 per cent of the 7-mm., 16 per cent of the6-mm., 25 per cent of the 5-mm., 36 per cent of the 4-mm., 49 per cent of the 3-mm., 64 per cent of the 2- mm. and 81 per cent of the 1-mm. Traversing the second and third rows of aperture the percentages will apply to the number of grairs left in passing over the preceding rows. Since the chance of 10-mm. grains passing through any number of apertures, no matter how great, is zero, this size is not tabulated. Where the calculation shows only a fraction of the volume of a whole grain eliminated it is assumed that a whole grain is eliminated where the fraction is greater than 0.5. A 10-mm. aperture is used for simplicity in computation. As an example of the computation the 9-mm. grain can be taken. Evidently in this case the area of the inner square is 1 sq. mm. and since the area of the aperture is 100 sq. mm., of the grains or 1 per cent are removed at each passage of a line of apertures.
1 See p. 246, et seq.